Specific Heat / Heat Capacity| Specific Heat / Heat Capacity |
In General
> s.a. heat; Heat Transfer.
* History: Einstein's 1907
article on the specific heat of solids introduced for the first time the
effect of lattice vibrations in the thermodynamic properties of crystals;
The next important step was the introduction of Debye's model.
* Heat capacity: The quantity
C:= ∂U/∂T, calculated either at constant
volume or at constant pressure, if appropriate; When the two are different,
Cp is greater than
CV because of the extra work
the system does in the expansion, but for a solid there is only one notion.
* Specific heat: The heat
capacity per mole c = C/n, with n the number
of moles, or per unit mass, c' = C/m.
> Online resources:
see Wikipedia page.
For Solids
* Dulong-Petit law: The
universality of specific heats of solids at high temperature, stating that
C = 5.94 cal/K per mole; It breaks down at low T (say, below
600 K or so, but it depends on the material), when equipartition no longer holds;
> s.a. energy; history of physics;
Wikipedia page.
* Einstein model: Atoms are
treated as non-interacting harmonic oscillators, so the phonon density of
states is a delta function at a single frequency; > s.a. Wikipedia
page.
* Debye model: The density
of states for atomic vibrations is modeled as g(ω)
= const ω2, up to some
ωH; This corresponds
to a constant speed of sound, and gives C proportional to
T 3; > s.a.
scienceworld page;
Wikipedia page.
@ References: Einstein AdP(07);
Shubin & Sunada mp/05 [geometric approach];
Grabowski et al PRB(09)
+ Grimvall Phy(09) [ab initio, up to melting point];
Mahmood et al AJP(11)nov [experimental determination];
González et al a1908 [Debye function].
Other Systems
> s.a. ising model; non-extensive statistics.
* Classical gas: From the
equipartition principle, CV =
(3/2) Nk [monatomic], (5/2) Nk or (5/2) Nk [diatomic].
* Liquids: A general theory of
the heat capacity of liquids has always remained elusive, in part because the
relevant interactions in a liquid are both strong and specific to that liquid;
2012, The "phonon theory of liquid thermodynamics" has successfully
predicted the heat capacity of 21 different liquids.
* Black hole: It is negative (as
is typical for a self-gravitating system, since there can be no equilibrium
with an infinite thermal bath), and given by
CS = T (∂S/∂T) = (∂M/∂T) = −8πM 2 = −TH2/8π .
@ Black hole: Gibbons & Perry PRS(78) [thermal Green's functions];
Górski & Mazur ht/97 [quantum effects, positive].
@ Boson system: Wang AJP(04)sep [above condensation T];
Ramakumar & Das PLA(06) [on a lattice].
@ Self-gravitating: Lynden-Bell & Wood MNRAS(67),
Lynden-Bell PhyA(99)cm/98-proc.
@ Other systems: Albuquerque et al PhyA(04) [quasi-periodic structures, oscillatory c(T)];
Moreira & Oliveira PRA(06)gq [relativistic particle on a cone];
Bolmatov et al SciRep(12)
+ news pw(12)jun [liquids].
Special Concepts and Results > s.a. sound [speed].
* Negative: In addition to
gravitating systems, it can happen in systems with small numbers of particles,
or some non-ergodic systems.
@ Negative:
Antoni et al proc(00)cm/99 [N-body];
Schmidt et al PRL(01)
+ pn(01)feb [Na clusters];
Thirring et al PRL(03) [non-ergodic];
Einarsson PLA(04)gq [conditions];
Posch & Thirring PRL(05) [and stellar stability];
Rao et al AP(08) [particles in box with potential well];
Staniscia et al PRL(10) [in the canonical statistical ensemble];
Serra et al EPL(13)-a1305 [finite quantum systems].
@ In non-extensive statistics:
Lenzi et al PLA(02);
Álvarez-Ramírez et al PLA(05).
@ Related topics: Pizarro et al AJP(96)jun;
Gearhart AJP(96)aug [and equipartition];
Filardo Bassalo et al NCB(01) [dissipative];
Fraundorf AJP(03)nov;
Behringer et al JPA(05) [microcanonical, finite size];
Starikov a1007 [from Bayesian approach].
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 26 aug 2019